Consumer Price Index

Consider an economy that produces just two goods-pizza and compact discs. The typical household in this economy currently spends 90% of its budget on pizza and the remaining 10% on CDs. If the price of pizza increases by 10% over the next year, and the price of CDs increases by just 2%, how would we measure the rate of inflation? One approach would be to take a simple average of the two price increases: ½ x 10% + ½ x 2% = 6%. Clearly, this would be a misleading indicator of the increased cost of living. Rather than a simple average, the Consumer Price Index uses what is known as a weighted average of the two price changes. In this example, the rate of inflation would be measured as 0.9 x 10% + 0.1x 2% = 9.2%. Conceptually, one can think of the economy as consisting of 10 items, 9 of which are pizza and one of which is CDs, then taking a simple average of these ten items. A simple average of n items is found by summing all the items and dividing by n, or equivalently, summing 1/n times each item: . A weighted average is similar, except each item is weighted by an amount w_i: , where the sum of the weights equals one. You will note that a simple average is a special case of the weighted average, where all of the weights are the same and equal to 1/n.

Mathematically, the CPI is computed as , where is the current price of good X_i, and is its price in the base year. The denominator is the sum of expenditures on a market basket of goods in the base year (year 0), while the numerator is the cost of this same market basket when evaluated at the prices now in effect (year 1). Alternatively, this could be written as the sum of n distinct terms, as follows: CPI = . Consider the first term in this series, . If we multiply both numerator and denominator by and rearrange, this can be written as . The term in parentheses measures the rate of price increase for good X₁ relative to the base year, while the first term measures the proportion of total base year expenditures accounted for by good X₁.

If we similarly multiply the numerator and denominator of each term in the summation by the appropriate price, we arrive at the following: CPI = where w_i = . This is the result we wanted: The CPI is a weighted average of the price increases of the individual goods, where each weight is the share of that good’s expenditures relative to total expenditures measured in the base year.